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10 DFT-based Green Function Approachfor Impurity Calculations

Rudolf ZellerInstitute for Advanced SimulationForschungszentrum Julich GmbH

Contents1 Introduction 2

2 Green function of the Kohn-Sham equation 2

3 Green function method for impurities 53.1 Density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Defect formation energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Forces and lattice relaxations . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 Long range perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5 Parameters for model Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . 16

4 Random Alloys and CPA 18

A Useful Green function properties 23

E. Pavarini, E. Koch, F. Anders, and M. JarrellCorrelated Electrons: From Models to MaterialsModeling and Simulation Vol. 2Forschungszentrum Julich, 2012, ISBN 978-3-89336-796-2http://www.cond-mat.de/events/correl12

http://www.cond-mat.de/events/correl12

10.2 Rudolf Zeller

1 Introduction

If one wants to apply density functional theory (DFT) to study the electronic structure of mate-rials locally perturbed by defect atoms or more generally of disordered dilute and concentratedalloys, the standard approach is to use band structure methods developed for periodic systems.The defect atoms are periodically repeated so that a periodic crystal with a large unit cell (su-percell) is obtained. The supercell must be large to minimize spurious interactions betweenthe defect atoms in adjacent supercells. Supercells with a few hundred atoms can be treatedroutinely today and sophisticated corrections for some of the unwanted interactions have beendeveloped. The situation was considerably different in the 1970s and 1980s when the comput-ing power was orders of magnitude smaller than today. Then Green function methods weremandatory to investigate the electronic structure of defect atoms more efficiently. The advan-tage of Green function methods is that the potential must be determined self-consistently onlyin the region where it noticeably differs from the one of the unperturbed host crystal. As a con-sequence Green function methods correctly describe the embedding of the local environment ofthe defect atoms in the otherwise unperturbed surrounding perfect crystal. It is the aim of thischapter to give an introduction into the concept of Green function methods for studying locallyperturbed crystals. For illustration of the concept and its advantages some key results will bepresented. These results were mainly obtained by a Green function (GF) technique which isbased on the Korringa-Kohn-Rostoker (KKR) band structure method [1, 2] and which was de-veloped in Julich over the last decades. This technique, the KKR-GF method, is particularlysuited for metallic systems, but can be applied also for semiconductors and insulators.

2 Green function of the Kohn-Sham equation

The basic quantity in density functional theory [3, 4] is the electronic density. The density canbe determined, if the Kohn-Sham single-particle equations[

−∇2r + veff(r)

]ϕi(r) = εiϕi(r) (1)

are solved.1 The normalized Kohn-Sham wavefunctions ϕi(r) can then be used to calculate thedensity by

n(r) = 2∑i

|ϕi(r)|2 (2)

where for a system with N electrons the sum is over the N/2 orbitals with the lowest values ofεi. The εi are the eigenvalues of the Hamiltonian H = −∇2

r + veff(r). The factor two accountsfor the assumed spin degeneracy. Alternatively to (2), the density can be calculated from theGreen function of the Kohn-Sham system. This Green function is defined as the solution of[

−∇2r + veff(r)− ε

]G(r, r′; ε) = −δ(r− r′) . (3)

1To simplify the notation, Rydberg atomic units ~2/2m = 1 are used throughout this chapter and usually theequations are given only for non-spin-polarized systems. The generalization to spin-polarized (magnetic) systemsis straightforward.

DFT Green Function Approach 10.3

Here the boundary condition G(r, r′; ε) → 0 for |r − r′| → ∞ is assumed and the symbol εdenotes a continuous complex variable in contrast to the real discrete variables εi. The resultfor the density is

n(r) = − 2

πIm

∫ EF

−∞G(r, r; ε)dε (4)

where the integral is understood as an integral in the complex ε plane on a contour infinitesi-mally above the real ε axis. The Fermi level EF is obtained by the condition that the densityn(r) integrated over all space gives the correct number of electrons. For instance, in neutralsystems, this number is determined by the sum of the nuclear charges. The result (4) can bederived as follows. In operator notation (3) can be written as

[H− ε]G = −I (5)

whereH is the Hamiltonian and I the unity operator. In terms of eigenvalues εi and eigenfunc-tions ϕi(r) the HamiltonianH can be expressed as

H =∑i

εiϕi(r)ϕ?i (r′) . (6)

This is true because the right hand of this equation acting on ϕi(r) leads to εiϕi(r) which is therequired result forHϕi(r). This follows from∑

j

εjϕj(r)

∫dr′ϕ?j(r

′)ϕi(r′) =

∑j

εjϕj(r)δij = εiϕi(r) (7)

where the orthonormality constraint∫dr′ϕ?j(r

′)ϕi(r′) = δij for the eigenfunctions was used.

By using that G is the inverse operator of ε−H the so called spectral representation

G(r, r′; ε) =∑i

ϕi(r)ϕ?i (r′)

ε− εi(8)

for the Green function is obtained. If this representation is inserted in (4) and the identity

limy→0+

1

x+ iy= P

1

x− iπδ(x) (9)

is applied to evaluate the imaginary part of (8), a delta function δ(ε − εi) appears which thencan be used to perform the integration in (4). The upper integration limit EF restricts the sumto eigenfunctions with εi ≤ EF and the equivalence of (2) and (4) is demonstrated.Equation (8) shows that the Green function has poles on the real ε axis at discrete values εi.These values represent the discrete part of the eigenvalue spectrum. In general, also a continu-ous part of the eigenvalue spectrum is possible. For instance in free space, with veff(r) = 0, theeigenfunctions are plane waves eikr with eigenvalues given by k2. These eigenvalues continu-ously cover the non-negative part of the real ε axis. Then (8) must be generalized into

G(r, r′; ε) =∑i

ϕi(r)ϕ?i (r′)

ε− εi+

∫ ∞−∞

ϕ(r; ε′)ϕ?(r′; ε′)

ε− ε′dε′ (10)

10.4 Rudolf Zeller

Fig. 1: Density of states for Ni (here only for majority electrons) and the corresponding quantityalong paths parallel to the real axis with distance 0.5, 1.8 and 4.7 eV. The picture is taken fromRef. [8]. It illustrates the increasing smoothness of the integrand with increasing distance fromthe real axis.

where the integral is over the continuous part of the spectrum. In periodic crystals the continu-ous part of the spectrum is realized by bands, for instance by the valence and conduction bandsin semiconductors. For the numerical evaluation of (4) it is important that the Green function isan analytical function2 of ε except for the singularities on the real axis. This means that the in-tegration (4) can be performed on a contour in the complex ε plane where the integrand is muchsmoother than just above the real axis as illustrated in Fig. 1. This procedure was suggested in anumber of papers [6–9] and leads to considerable savings of computing time. Usually 30 to 40points in the complex ε plane are enough for accurate evaluations of (4) provided that the pointsare chosen dense enough near EF , where the contour necessarily approaches the real axis.An important quantity, which is often used to provide an understanding of the electronic struc-ture of materials in a single-particle picture, is the local density of states within in a volume V.It is defined as

nV (ε) = 2∑i

δ(εi − ε)∫V

dr |ϕi(r)|2 (11)

and gives the distribution of occupied and unoccupied electronic states within the volume V ,for instance the local volume corresponding to one atom in the system. In terms of the Greenfunction the local density of states is given by

nV (ε) = −2

πIm

∫V

drG(r, r; ε) (12)

as can be verified by using the spectral representation (8). The total density of states is obtainedif the integrals in (11) or (12) are done over all space.

2For an elementary introduction to classical Green functions and their analytical properties the textbook ofEconomou [5] is a good source.


3 Green function method for impurities

Historically the Green function method is attributed to Koster and Slater [10] who expanded theGreen function in terms of Wannier functions. Because of the difficult construction of Wannierfunctions at that time the capabilities of this approach were rather limited. At the end of the1970s Green function methods for the calculation of the electronic structure of impurities insolids received new attention. Inspired by the success of DFT calculations for periodic solids,several groups invested a considerable amount of work into the development of Green functionmethods for impurity calculations. They used a number of different techniques to calculate thethe Green function of the periodic host crystal which is needed for the subsequent impuritycalculations. The LCAO-GF method [11,12] was based on an expansion in linear combinationsatomic orbitals, the LCGO-GF method [13] on an expansion in linear combination of Gaussianorbitals, the LMTO-GF method [14,15] on the linear muffin-tin orbital method and the KKR-GFmethod [16, 17] on the Korringa-Kohn-Rostoker band structure method.In the beginning the Green functions were calculated mostly by using the spectral representa-tion (8). This is easy for the imaginary part of the Green function because according to (9) theimaginary part of the denominator ε−εi leads to a delta function so that only εi values contributewhich are in the ε range for which the Green function is needed. From the imaginary part thereal part was then obtained by the Kramers-Kronig relation

ReG(r, r′; ε) = − 1

πP

∫ ∞−∞

dε′1

ε− ε′ImG(r, r′; ε′) . (13)

Here in principle, the imaginary part is needed along the entire real ε axis. In practice, ap-proximations were used, either the imaginary part in the integrand was neglected for higher εvalues or it was replaced by an analytical approximation [18]. Another possibility is to use basisfunctions in a finite Hilbert space [15].The problem that an infinite number of eigenstates contributes in (8) can be avoided if the hostGreen function is directly calculated by Brillouin zone integrations. This technique, which di-rectly calculates the Green function for a given energy, is used in the KKR coherent potentialapproximation (KKR-CPA) [19, 20] and has been implemented also in the KKR-GF method.This procedure is perhaps most easily understood in terms of reference Green functions. In-stead of using the defining differential equation (3) or the spectral representation (8), the Greenfunction is calculated from the integral equation

G(r, r′; ε) = Gr(r, r′; ε) +

∫dr′′Gr(r, r′′; ε)∆v(r′′)G(r′′, r′; ε) . (14)

Here Gr is the Green function of a suitably chosen reference system with reference potentialvr(r) and ∆v(r) = veff(r)− vr(r) is the perturbation of the potential given by the difference ofthe Kohn-Sham potential and the reference potential. The Green function Gr is determined by[

−∇2r + vr(r)− ε

]Gr(r, r′; ε) = −δ(r− r′) . (15)

The calculation of Green functions by using reference systems is a powerful concept for thecalculation of the electronic structure for systems with a complicated geometric structure. An

10.6 Rudolf Zeller

Fig. 2: Illustration for the geometry of one impurity atom in an atomic chain which consists ofother atoms than the otherwise periodically repeated chains at the step edges of a vicinal (711)surface of an fcc crystal.

example for such a system, shown in Fig. 2, is a vicinal (711) surface of a face-centered-cubiccrystal decorated with adatoms at step edges and within step edges. The Green function for thissystem can be constructed by the successive calculation of Green functions of simpler systems.i) Starting from free space the Green function for the bulk crystal is calculated treating the fullbulk potential as perturbation using periodicity in three dimensions. ii) Several layers of the bulkcrystal are removed, the removed potential is treated as perturbation using the two-dimensionalsurface periodicity. As a consequence of the fact the electrons cannot tunnel through severalempty layers one obtains a slab which is decoupled from the rest of the bulk crystal. iii) Atomicchains are added at step edges. The potential perturbation is periodic in two dimension, butlocally restricted to the vicinity of the step edges. iv) A chain of different atoms is inserted,only one-dimensional periodicity along the considered step edge is preserved, but the potentialperturbation is confined to the vicinity of the considered chain. v) Finally, an impurity atom isinserted, periodicity is fully lost, but the potential perturbation is essentially confined to atomsin the vicinity of the impurity.

During this successive construction, periodicity can be used in the dimensions where it existswhile in the remaining dimensions the potential perturbation is localized in the vicinity of thereplaced atoms. In each successive step the integral equation must be solved only within aregion of space where the potential differs non-negligibly from the one of the reference system.Once the Green function has been obtained for r and r′ in this region, the Green function in allspace can be obtained simply by multiplications and integrations. This represents an enormousadvantage of Green function methods over supercell methods because the size of the regionfor which the integral equation must be solved is determined by the extent of the potentialperturbation and not by the usually much larger extent of the perturbed wavefunctions or Greenfunctions.


Fig. 3: Partitioning of space and illustration for the definition of Rn and Rn′, which define the

positions of the cell centers, and r and r′, which are vectors within the cells.

An effective way to solve the integral equation (14) is provided by the KKR-GF method whichdoes not rely on the determination of eigenvalues and eigenfunctions with their orthonormalityconstraints, but uses ideas of multiple-scattering theory. In this theory space is divided intonon-overlapping regions, for instance cells around each atom, and the calculation of the Greenfunction is broken up into two parts. First, single-scattering quantities, which depend only onthe potential in a single cell, are determined. Second, the multiple-scattering problem is solvedto obtain the correct combination of all single-scattering events.The mathematical basis for the KKR-GF method is the fact that the Green function for freespace, which is given by

G0(r, r′; ε) = − 1

4π

exp(i√ε|r− r′|)

|r− r′|, (16)

can be written in cell-centered coordinates as

G0(r+Rn, r′ +Rn′; ε) = δnn′G0(r, r′; ε) +

∑LL′

JL(r; ε)G0,nn′

LL′ (ε)JL′(r′; ε) (17)

with analytically known Green function matrix elements G0,nn′

LL′ (ε) which do not depend on theradial coordinates r or r′. Here Rn and Rn′ are the coordinates of the cell centers, usually theatomic positions, and r and r′ are coordinates within the cells which originate at the cell centers.An illustration for these coordinates is given in Fig. 3. The functions

JL(r; ε) = jl(r√ε)Ylm(r) (18)

are products of spherical harmonics Ylm with spherical Bessel functions. Angular variables aredenoted by r = r/r and radial variables by r = |r|. The symbol L is used as compact notationfor the angular momentum indices l and m and sums over L denote double sums over l and|m| ≤ l.

10.8 Rudolf Zeller

If one proceeds as in [21], it is straightforward to show that the solution of (14) is given by

G(r+Rn, r′ +Rn′; ε) = δnn′Gn

s (r, r′; ε) +

∑LL′

RnL(r; ε)G

nn′

LL′(ε)Rn′

L′(r′; ε) . (19)

Here Gns is the single-site Green function for cell n which satisfies the integral equation

Gns (r, r

′; ε) = Gr(r, r′; ε) +

∫n

dr′′Gr(r, r′′; ε)∆vn(r′′)Gns (r′′, r′; ε) (20)

and RnL are the single-site solutions which satisfy the integral equations

RnL(r; ε) = JL(r; ε) +

∫n

dr′Gr(r, r′; ε)∆vn(r′)RnL(r′; ε) . (21)

In (20-21) the integration is only over the volume of cell n and ∆vn(r) = ∆v(r + Rn) is thepotential perturbation in this cell n. The Green function matrix elements Gnn′

LL′ used in (19)satisfy the matrix equation

Gnn′

LL′(ε) = Gr,nn′

LL′ (ε) +∑n′′

∑L′′L′′′

Gr,nn′′

LL′′ (ε)∆tn′′

L′′L′′′(ε)Gn′′n′

L′′′L′(ε) (22)

which computationally represents a linear algebra problem. The difference ∆tn of the so-calledsingle-site t matrices is given by

∆tnLL′(ε) =

∫n

dr JL(r; ε)∆vn(r)Rn

L′(r; ε) . (23)

Together (19-23) provide a computationally convenient solution of the integral equation (14)without the need to determine eigenvalues and eigenfunctions. The only real approximationwhich must be made is the truncation of the infinite sums over L to a finite number of terms.This determines the angular momentum cutoff lmax used in the KKR-GF method. Usuallylmax = 3 or lmax = 4 is sufficient for accurate results.If the density of states integrated over all space is needed, for instance for the total energycalculations, it is important that the integration in (12) can be performed analytically if thevolume V extends over all space. This is the essence of Lloyd’s formula [22] which gives thedifference of the integrated density of states

∆N(ε) =

∫ ε

dε′∆n(ε′) (24)

of the perturbed and unperturbed systems by logarithms of determinants. The formula can bewritten as

∆N(ε) =1

πIm∑n

∆ ln det |αnLL′(ε)| −1

πIm ln det |δnn′

LL′ −∑L′

Gr,nn′

LL′ (ε)∆tn′

L′L(ε)| (25)

which is the KKR equivalent of the operator identity (61) derived in the appendix. The matrixα in (25) is defined by [23]

αnLL′(ε) = δLL′ +

∫n

drHL(r; ε)vn(r)Rn

L′(r; ε) (26)

for the potential vn(r) and analogously for the reference potential, where the functions

HL(r; ε) = h(1)l (r√ε)Ylm(r) (27)

are products of spherical harmonics with spherical Hankel functions of the first kind.


Fig. 4: Local density of states (within the impurity cell) for Mn impurities in Cu and Ag asfunction of energy (in eV) relative to the Fermi level. The minority spin density of states isplotted downwards so that it does not overlap with the majority one. The dotted curves showthe integrated density of states. The picture is taken from Ref. [25].

3.1 Density of states

Although the single-particle eigenvalues εi and the density of states n(ε) calculated in densityfunctional theory are only formal mathematical quantities, they often agree qualitatively withexperiment and even more often they are used to provide a single-particle understanding of thechemical and physical mechanisms which lead to the observed behaviour of materials. Amongthe first systems studied by the KKR-GF method were 3d transition metal impurity atoms innoble metals which are the classical systems considered by Anderson in his famous paper onthe Anderson impurity model [24].In Fig. 4 the local density of states calculated with the local density approximation (LDA) ofDFT is shown for Mn impurities in Cu and Ag. The differences for the two spin directionsare caused by the fact that transition metal atoms of the Fe series can gain magnetic exchangeenergy by spin alignment. Anderson has discussed this behaviour in terms of Lorentzian typevirtual bound states. Fig. 4 shows that these virtual bound states are reproduced in LDA-DFTcalculations. In particular, the minority density of states shows almost Lorentzian type featureswith peak positions just above the Fermi level. However, also distortions from the Lorentzianform are clearly visible. They are caused by band structure effects arising from hybridization ofthe 3d-states of the impurity atoms with the host d-states. Interestingly, in 1980 when Ref. [25]was published, it was seriously doubted that the observed splitting of the peak positions wasthe correct density functional answer, because conflicting results with much smaller splittingexisted [26]. One criticism was that the observed spin splitting could be an artifact of the factthat only a potential perturbation in the impurity cell was used. Later calculations [27], wherepotential perturbations were also used in neighboring cells, confirmed the results shown inFig. 4. Moreover, measurements [28,29] with various electron-spectroscopy techniques showedsimilar spin splittings as calculated. Fully quantitative agreement between calculations andexperiment, of course, cannot be expected because states calculated with LDA are not to beidentified as measurable quantities.

10.10 Rudolf Zeller

3.2 Defect formation energies

An important problem in the study of defects is the calculation of total energy differences whichare the basic quantities necessary for understanding the microscopic origin of the formation ofalloys and many other physical processes such as diffusion, short-range ordering, segregation.This is particularly true for vacancy formation and migration energies, which are difficult tomeasure, but important quantities to understand the thermodynamic and kinetic behaviour ofmetals and semiconductors. In supercell methods these numbers are difficult to calculate asexpressed, for instance, in the following sentences taken from Puska et al. [30], who used su-percells with up to 216 atoms. These authors write in the introduction The vacancy in Si canbe considered as the simplest example of a point defect in a semiconductor lattice and in theconclusions The convergence of the results is shown to be very slow and If the supercell is notlarge enough the long-range ionic relaxation pattern, especially in the [110] zigzag direction,may not be properly described. Since then the vacancy in Si has been reinvestigated severaltimes by supercell calculations, where due to the ever increasing computer power the supercellsize has gradually increased. However, the convergence problem remains as stated in one of themost recent articles by Corsetti and Mostofi [31]. These authors who used supercells with up to1000 atoms write in the conclusions Our calculations confirm the slow finite size convergenceof defect formation energies and transition levels, ... They also argue that future increase ofsupercell sizes can reduce the spurious interactions between the vacancies in different cells, butthat then another problem must be considered seriously. The problem is that total energy differ-ences due to defect atoms are not calculated directly, but are obtained by numerical subtractionof supercell energies with and without defect. With increasing supercell size the numbers to besubtracted become larger which puts heavy demands on the numerical precision.

In contrast to supercell methods, Green function calculations are not plagued by these problems.The formation energy is calculated directly, not by energy differences, spurious interactionsdo not exist, and the region in space where the self-consistent calculations must be done isdetermined by the range of the potential perturbation and not by the usually much larger rangeof wavefunctions or Green function perturbations. Actually, the Si vacancy was consideredas one of the earliest systems in Green function impurity calculations [11, 12]. At that time,however, it was not possible to calculate accurate total energies. Green function methods usingbasis sets suffered from insufficiently accurate basis functions and the KKR-GF method fromthe spherical approximation for the potential used in each cell. Only at the beginning of the1990s accurate total energy calculations became possible, both due to the increased computingpower and even more so because of the development of advanced numerical techniques. Resultsfor some vacancy formation energies calculated by the KKR-GF method [32] are shown inTable 1. Compared to the original publication [33] the values for Cu and Ni contain correctionsof about -0.04 and -0.08 eV which are energy gains obtained if the atoms relax to the correctequilibrium positions.

For the calculation of defect formation energies it is very important that Lloyd’s formula is usedto obtain the single-particle contribution. Within density functional theory the total energy con-


Table 1: Calculated and experimental values for the vacancy formation energy of selectedtransition metals (in eV). The calculations for Cu and Ni include the effect of lattice relaxations.

Cu Ag Ni PdTheory 1.37 1.20 1.68 1.57Experiment 1.28 1.11 1.79, 1.63 1.85, 1.54

Table 2: Solution energy (in eV) for a V impurity in Cu calculated using potential perturbationsin the vanadium cell and in different numbers of shells of surrounding Cu neighbors.

shells 0 1 2 3 4cells 1 13 19 43 55ELloyd 1.44 0.73 0.73 0.73 0.73Elocal 1.60 1.93 1.38 0.75 0.52

sists of a sum of the kinetic energy T [n(r)], the electrostatic Hartree energy and the exchangecorrelation energy. The kinetic energy is usually evaluated as

T [n(r)] =∑i

εi −∫

drn(r)veff(r) =

∫ EF

dε ε n(ε)−∫

drn(r)veff(r) (28)

= EFN(EF )−∫ EF

dεN(ε)−∫

drn(r)veff(r)

by using the single-particle energies εi, where the last result is obtained from integration by partsover ε. If the density of states n(ε) or the integrated density of states N(ε) is calculated fromthe Green function by (11) and (24), an explicit summation over all cells of the infinite crystalis required. This problem is avoided if Lloyd’s formula (25) is used which already contains theintegration over all space.The different behaviour of the shell convergence of energies calculated with and without Lloyd’sformula is illustrated in Table 2 where results for the solution energy for a vanadium impurityin copper are shown. While the use of Lloyd’s formula leads to a converged result already for13 cells, the use of (11) with summation over cells only gives poor results. This behaviourreflects the fact that wavefunction or Green function perturbations are much longer ranged thanthe potential perturbation. Density changes, which contribute, for instance, in the last term of(28) and in other parts of the energy functional, exist outside of the perturbed potential region,but these changes are unimportant because of the variational properties of the energy functional.

3.3 Forces and lattice relaxations

Substitutional impurities, in general, have a different size than the host atoms. This causes dis-placements of the neighboring atoms away from their ideal positions which they occupy in theunperturbed host crystal. In metals, because of the high coordination number, the displacements

10.12 Rudolf Zeller

are rather small and can be neglected in the calculations of many physical properties. However,sometimes the displacements have significant effects and their size and direction should becalculated. In electronic structure methods displacements are calculated usually from the con-dition that the forces on the atoms should vanish and the forces are determined usually by theHellmann-Feynman theorem

Fn = − ∂E

∂Rn

∣∣∣∣n(r;Rn)

−∫

drδE

δn(r)

∂n(r;Rn)

∂Rn(29)

which means that the force Fn on atom n is determined by the derivative of the total energy Ewith respect to the coordinate Rn of atom n. Here the first term, evaluated at constant densityn(r;Rn), is the Hellmann-Feynman (HF) force and the second term is necessary if approxima-tions are made in the solution of the Kohn-Sham equations. The second term vanishes in anexact treatment, because then δE

δn(r)= EF is constant and because the total number of electrons

Nel =∫drn(r;Rn) does not depend on the atomic positions. Within a full potential KKR

formalism, the Kohn-Sham equations for the valence electrons are solved rather accurately, theonly approximation is the lmax cutoff. This means that the second term usually contains a neg-ligible contribution from the valence electrons. For the core electrons, however, this term givesa considerable contribution if the core states are calculated as usual in an atomic fashion usingonly the spherical part of the potential. With a spherical ansatz ncore for the core density, theresulting expression for the force is

Fn = Zn ∂VM(r)

∂r

∣∣∣∣r=Rn

−∫d3r ncore(|r−Rn|) ∂veff(r)

∂r(30)

whereZn is the nuclear charge, VM(r) the Madelung potential and veff(r) the Kohn-Sham poten-tial. Due to the vector character of the potential derivatives in (30), only the l = 1 componentsof VM and veff are needed. Since these quantities are anyhow calculated in a full-potential KKRtreatment, the calculation of forces is easy. There are also no Pulay corrections [34] requiredwhich arise in basis set methods if the basis functions depend on the atomic positions.While force calculation are simple, calculations of atomic displacements, which occur if theimpurity atoms are smaller or larger than the host atoms, are more complicated in the KKRmethod. The main reason is the site-centered angular momentum expansion used in the Greenfunction expression (19) which is needed around the displaced sites. While original undisplacedand new displaced sites can be used together in the calculation of the Green function with appro-priate artificial zero potentials on non-participating sites, it is numerically simpler to transformthe Green function matrix elements from undisplaced to displaced sites [35, 36]. The referenceGreen function matrix elements are transformed by

Gr,nn′

LL′ (ε) =∑L′′L′′′

ULL′′(sn; ε)Gr,nn′

L′′L′′′(ε)UL′L′′′(sn′; ε) (31)

where sn is the shift of atom at site Rn to the new site Rn + sn. The transformation matrix isgiven by

UL′L(sn; ε) = 4π

∑L′′

il′+l′′−lCLL′L′′jl′′(

√εsn)YL′′ (sn) (32)


−1.5

−1.0

−0.5

0

0.5

1.0

1.5

2.0

2.5

Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge

Displacem

entoffirstN

N(%)

EXAFS

Fig. 5: Displacements of the nearest Cu neighbors around impurities from Ti to Ge in Cu. Thedisplacements are given in percent of the nearest neighbor distance between the atoms in un-perturbed Cu. Experimental results obtained by EXAFS measurements are shown by triangleswith error bars. The picture is taken from Ref. [37].

where CLL′L′′ =∫

4πdrYL(r)YL′(r)YL′′(r) are Gaunt coefficients, jl spherical Bessel functions

and YL spherical harmonics. Together with a similar transformation for the t matrix, the follow-ing algebraic Dyson equation

G = Gr + Gr[t− tr]G (33)

must be solved. While the transformation (31) is exact, if the sums over L′′ and L′′′ are extendedover infinite angular momenta, in practical calculations these sums must be truncated. For largedisplacements a relatively high lmax value is needed, but lmax = 4 is sufficient for displacementsup to 10 % of the nearest neighbor distance as they occur around the substitutional impuritiesconsidered here.Fig. 5 shows calculated displacements of nearest neighbor Cu atoms around impurities fromthe 3rd series of the periodic table together with experimental data derived from extended x-rayabsorption fine structure (EXAFS) measurements. Most of the impurities lead to an outwarddisplacement of the Cu neighbors which means that these impurities are bigger than Cu. OnlyNi and Co are smaller with inward displacements. The displacements induced by Fe are verysmall and for a Cu impurity they are, of course, zero.In contrast to metals, lattice relaxations in semiconductors are usually considerably larger be-cause of more open structures and lower coordination numbers. For the defect pairs shown inFig. 6 the displacements reach up to 10 % of the nearest neighbor distance which is about thelimit for the U transformations. The figure shows a comparison of displacements calculated fordonor-acceptor pairs in Si by the full-potential KKR-GF method and a pseudopotential ab-initiomolecular dynamics program applied to a supercell with 64 atoms [38]. The displaced positions

10.14 Rudolf Zeller

Si

Si

Si

Si

Si

Si

In

In

In

AsSb

P

7.2 (7.5)%0.8 (1.1) %

8.3 (8.2)%14.1 (13.4)%

6.5 (7.0)%

6.7 (6.5)%

2.6 (2.8)%5.3 (5.1)%

1.8 (0.6)%

4.8 (4.5)%

1.0 (1.3)%

6.1 (5.7)%

Si

D

In

Si

Fig. 6: Calculated displacements for InSb, InP and InAs defect pairs in Si. The results are givenin percent of the nearest neighbor distance between the bulk Si atoms. Two sets of numbers aregiven. The number in parentheses have been calculated by a pseudopotential method, the othernumber by the KKR-GF method.

obtained by the two methods are essentially the same, but the KKR-GF method can give moreinformation, in particular for properties that are determined by the core electrons, like hyperfinefields or electric field gradients.

The considered defect pairs are electrically and magnetically inactive and experimental infor-mation about the structure is difficult to obtain. One of the few methods to investigate suchdefects are perturbed angular correlation (PAC) experiments which measure the electric fieldgradients. The calculated electric field gradients depend sensitively on the lattice relaxations ofthe defect complex. While calculations without lattice relaxations give the wrong trend with re-spect to the atomic numbers of the donor atoms, the agreement greatly improves, if the relaxedconfigurations are considered [39], as for instance given in Fig. 6 for donor-acceptor pairs inSi. Thus a reliable calculation of the relaxations is crucial for understanding the electric fieldgradients.

Efficient and accurate force calculations open the possibility to study phonon dispersion rela-tions. Within a Green function impurity method the calculations can be done in real space bydirectly determining the Born-von Karman force constants according to their definition. Oneatom is displaced by a finite amount and the induced forces on all atoms are calculated. Fouriertransformation gives the dynamical matrix and phonon frequencies and eigenstates in the Bril-louin zone are easily obtained. For cubic crystals a single self-consistent calculation is enoughto determine the whole phonon spectrum. Fig. 7 shows the phonon dispersion curves of Alcalculated in this way by the KKR-GF method. As can be seen a calculation including 6th near-


0

1

2

3

4

5

6

7

8

9

10

11

Frequency(THz)

Γ Χ Χ Γ L[00ζ] [0ζ1] [0ζζ] [ζζζ]

Al

Fig. 7: Calculated phonon spectrum (continuous lines) of fcc Al and experimental results (fulldots). The central Al atom is shifted by 0.5% and the forces on six shells of neighboring atomsare calculated self-consistently, yielding force constant parameters for six nearest neighborshells. The figure is taken from Ref. [37].

est neighbor interactions reproduces the experimental data quite well. A disadvantage of thisapproach is that a relatively big cluster, including many atoms, must be used to account for longrange elastic interactions present for example in semiconductors.

3.4 Long range perturbations

Magnetic impurities in non-magnetic materials induce magnetic polarization oscillations on thesurrounding host atoms. Usually these oscillations are small effects. Typically a Cu atom asnearest neighbor of an impurity from the 3d series carries an induced moment of about 10−2µB

and the size of the moments decreases with increasing distance from the impurity as the thirdpower of distance. Despite of this, as discussed in Ref. [40], very nice experimental informationabout the magnetization oscillations in Cu exists due to measurements of Knight shift satellitesby the Slichter group. The measured Knight shift satellites can be related to calculated hyperfinefields as detailed in Ref. [40]. Fig. 8 shows calculated hyperfine fields on twelve shells ofCu neighbors around a Mn impurity, which means that potentials for 225 atoms have beencalculated self-consistently. The corresponding experimental results derived from the Knightshift measurement are shown by red squares. Calculation and experiment nicely agree if thecorrect assignment of the experimentally observed peaks to the different shells has been made.While the assignment of calculated hyperfine fields to the different shells is unambiguous, theassignment of experimental peaks is more difficult. It relies on the symmetry of the shells, onthe intensity of the peaks arising from the number of atoms in the shells and on the magnitudeof the derived hyperfine field according to the rapid decrease with distance.

10.16 Rudolf Zeller

Fig. 8: Hyperfine fields of Cu atoms in different shells around a Mn impurity in Cu. Thecalculated values (solid curve) were multiplied with (Rn/a)

3, where Rn is the distance from theMn nucleus and a the lattice constant of Cu. The numbers indicate the different shells and theexperimental values are shown as red squares. The picture is taken from Ref. [41].

3.5 Parameters for model Hamiltonians

Green function impurity calculations can be used to obtain parameters for model Hamiltoni-ans if constraints are applied in density functional theory [42]. Constraints are already usedin the formal development of density functional theory, for instance the density is constrainedto give the correct number of electrons and the Kohn-Sham orbitals must be normalized as∑

α(ϕαi , ϕ

αi ) = 1. Another example is Levy’s [43] explicit definition of the energy by con-

strained minimization over all many-electron wavefunctions which give the same density. Theidea of constrained density-functional theory [42] is the extension to quite arbitrary constraints.This idea is useful if one wants to calculate total energy differences which depend on a parame-ter, for instance on NV , the number of electrons inside a volume V . The constraint can be takeninto account by modifying the energy functional E[n(r)] into

E[n(r)] = E[n(r)] + v

[NV −

∫V

n(r)dr

](34)

where the constraint is guaranteed by the Lagrange parameter v. The minimization of (34)with respect to n(r) leads to an additional potential v in the Kohn-Sham equations, which isconstant and only acts in V and is zero elsewhere. This potential must be adjusted such thatthe resulting density n(r) gives exactly NV electrons in volume V . Instead of calculating theenergy differences from the functional E[n(r)] by subtracting the energies for different valuesof the parameter NV , it is computationally easier to calculate the difference directly by theHellmann-Feynman theorem

dE(NV )

dNV

= v ⇒ ∆E(NV ) =

∫ NV

N0

v(N ′) dN ′ (35)

which only requires the knowledge of the potential v(N ′). Physically, the potential v can beviewed as the force necessary to constrain the system to the desired state and ∆E as the strainenergy of the system.


Fig. 9: Constraining field (left) and energy difference (right) as function of the deviation of theoccupation number Nf from its ground state value N0

f . Pictures are taken from Ref. [42].

Instead of varying the total charge it is more interesting to vary partial charges, for instancethe number of d- or f -electrons in the impurity cell. Minimization then leads to a constantprojection potential v(Nd,f ) which acts only on states with l = 2 or l = 3 character in theconsidered cell. The energy differences obtained in this way can be related to the screenedCoulomb parameters Ud and Uf (Hubbard U), for instance as

Uf = ∆E(N0f + 1) +∆E(N0

f − 1) (36)

where N0f is the number of f -electrons in equilibrium. In the calculation the screening is pro-

vided by the the s- and p-states which can adjust to the changed number of d- or f -electrons.An early application of constrained density functional theory was the calculation of the Coulombparameter Uf for Ce impurities in Ag and Pd [42]. The number Nf of f -electrons was deter-mined by integrating the local density of states (12) using only l = 3 angular momentumcomponents for n(ε). The ground state values of Nf were determined as N0

f = 1.18 for Ce inPd and as N0

f = 1.25 for Ce in Ag. A constraining projection potential vf was applied withinthe muffin-tin sphere around the Ce impurity and the dependence of Nf on vf was determined.Both the constraining potential and the energy difference are plotted in Fig. 9. The dependenceof vf on Nf is almost linear except for Pd near ∆Nf = −1, where it becomes increasinglydifficult to remove all f -states due to their strong hybridization with d-states of Pd. The en-ergy differences depend almost quadratically on ∆Nf and the screened Coulomb parametersaccording to (36) turned out be 6.6 eV for Ce in Ag and 8.1 eV in Pd.Another early application of constrained density functional theory was the calculation of in-teraction energy differences between the ferromagnetic and antiferromagnetic configuration ofimpurity pairs in metals [44]. In these calculations the local magnetic moment of one of theimpurities is constrained to an arbitrary value M and the lowest energy compatible with theconstraint is determined by a modified functional

E[n(r),m(r)] = E[n(r),m(r)] +H

[M −

∫V

m(r)dr

]

10.18 Rudolf Zeller

Fig. 10: Difference of the magnetic interaction energy difference ∆E(M) and the constrainingmagnetic field H(M) as function of the prescribed local impurity moment. The results are forpairs of Mn and Fe impurities on nearest neighbor sites in a Cu crystal (from Ref. [44]).

where the Lagrange parameterH is a constraining longitudinal magnetic field, which is constantin the cell of one impurity with volume V and zero elsewhere. This field is chosen such that theintegral of the magnetization m(r) over the cell gives the desired value of the moment.Similar to (35) the energy difference is given by

∆E(M) =

∫ M

M0

H(M ′)dM ′

where M0 is the value of M in the reference state. For instance in Fig. 10, the reference state isthe antiferromagnetic configuration, for which the moments for the two impurities have oppositesign. This state corresponds to the left minima of the ∆E(M) curves in Fig. 10, while the rightminima correspond to the ferromagnetic configuration. Both configurations are stable as theenergy minima with vanishing constraining field H indicate. The energy differences betweendifferent magnetic configurations determine exchange parameters Jij which may be used in aHeisenberg model

H = −∑i,j

JijSiSj ,

which can be treated much faster for large and complex systems than fully self-consistent spindensity functional calculations.

4 Random Alloys and CPA

For real alloys the consideration of one or even two impurities is relevant only for the very di-lute limit where interactions between impurity clusters can safely be neglected. For increasingimpurity concentration these interactions become more and more important. Instead of calcu-lating the Green function for a single configuration realized in a particular system, one is theninterested in spatial or configurational averages 〈G〉 of the Green function over many differentpossible configurations [45]. For the derivation of expressions for the averaged Green function


〈G〉 it is convenient to consider first the single-site problem and then the interaction betweenthe different sites. The single-site problem can be written in operator notation as

Gα = g + gVαGα (37)

where all quantities are considered as integral operators3 and g and Vα are used as shorternotations for Gr and ∆vn. In order to emphasize that in this section single-site quantities areintegral operators, the sites are labelled here by Greek letters. With the definition tαg = VαGα

the last equation can be solved asGα = g + gtαg . (38)

Multiplication with Vα from the left leads to

VαGα = tαg = Vαg + Vαgtαg . (39)

This is valid if tα is determined by

tα = Vα + Vαgtα . (40)

Using spatial variables the last equation can be written as

tα(r, r′) = Vα(r)δ(r− r′) +

∫dr′′ Vα(r)g(r, r

′′)tα(r′′, r′) (41)

where the potential operator is local as characterized by the delta function. By iterating (40) astα = Vα + VαgVα + VαgVαgVα . . . one sees that tα is a single-site quantity. It is only needed incell α because all terms contain factors Vα on the left and right and Vα is restricted to cell α.For the multiple-site problem, where the potential perturbation is given by V =

∑α Vα, it is

convenient to introduce an operator F which connects G and its average 〈G〉 by

G = F 〈G〉 . (42)

If this is used in G = g + gV G, which is (14) in operator notation, the result is

F 〈G〉 = g + gV F 〈G〉 (43)

Averaging gives〈G〉 = g + g〈V F 〉〈G〉 (44)

where 〈F 〉 = 1 was used which follows from (42). The last equation contains only averagedquantities and could be solved easily if 〈V F 〉 is known.4

The task is now to derive an equation for 〈V F 〉. Subtraction of (43) and (44) and omitting thecommon operator 〈G〉 on both sides of the resulting equation gives

F − 1 = g(V F − 〈V F 〉) . (45)

3Integral operators, for instance g, act on arbitrary functions f(r) as∫dr′g(r, r′)f(r′).

4 The quantity 〈V F 〉 is usually called self-energy and labelled by the letter Σ.

10.20 Rudolf Zeller

which multiplied with V leads to

V F = V + V g(V F − 〈V F 〉) (46)

This is the basic result used for further considerations. Now the crucial point is that V F can bewritten as a sum of single-site terms. For that purpose operators fα are defined by the equationVαF = tαfα. With V =

∑α Vα one obtains V F =

∑α VαF =

∑α tαfα and (46) can be

written as ∑α

tαfα =∑α

Vα +∑α

∑β

Vαgtβfβ −∑α

∑β

Vαg〈tβfβ〉 . (47)

On the other hand, if (40) is multiplied with fα and summed over α, the result is∑α

tαfα =∑α

Vαfα +∑α

Vαgtαfα . (48)

Subtraction of (47) and (48) gives

0 =∑α

Vα(fα − 1−∑β 6=α

gtβfβ +∑β

g〈tβfβ〉) (49)

where the restriction β 6= α arises from the subtraction of the last term of (48). The last equationis satisfied if fα is determined by the implicit equation

fα = 1 +∑β 6=α

gtβfβ −∑β

g〈tβfβ〉 . (50)

In general, this equation cannot be solved exactly, but it can be used to obtain approximations.Iteration of (50) starting with fα = 0 leads to

fα = 1 (51)

+∑β 6=α

gtβ −∑β

g〈tβ〉

+∑β 6=α

∑γ 6=β

gtβgtγ −∑β 6=α

∑γ

gtβg〈tγ〉 −∑β

∑γ 6=β

g〈tβgtγ〉+∑β

∑γ

g〈tβ〉g〈tγ〉

+ . . .

where terms containing three or more t operators are not shown. In order to obtain approxima-tions for 〈V F 〉 the last equation is multiplied with tα, summed over α and averaged. This leadsto

〈V F 〉 =∑α

〈tα〉 (52)

+∑α

∑β 6=α

〈tαgtβ〉 −∑α

∑β

〈tα〉g〈tβ〉

+∑α

∑β 6=α

∑γ 6=β

〈tαgtβgtγ〉 −∑α

∑β 6=α

∑γ

〈tαgtβ〉g〈tγ〉

−∑α

∑β

∑γ 6=β

〈tα〉g〈tβgtγ〉+∑α

∑β

∑γ

〈tα〉g〈tβ〉g〈tγ〉+ . . . .


Here it is important to note that this expression is still exact although only terms up to thirdorder in tα are written down.Now approximations can be made. An important approximation, which leads to a considerablesimplification, is the neglect of correlations between different sites. While electronic structurecalculations for alloys nowadays do not necessarily need this approximation, for instance in thenon-local CPA of Rowlands et al. [46, 47], these advanced techniques are difficult to discussand will not be considered here. The neglect of correlations means that averages 〈tαgtβ〉 can befactorized as 〈tα〉〈tβ〉 if the sites α and β are different. This leads to

〈V F 〉 =∑α

〈tα〉 −∑α

〈tα〉g〈tα〉 (53)

+∑α

〈tα〉g〈tα〉g〈tα〉+∑α

∑β 6=α

〈tαg〈tβ〉gtα〉 −∑α

∑β 6=α

〈tα〉g〈tβ〉g〈tα〉 − . . . .

From this expression a number of approximations can be derived which are far better than themost simple approximation which replaces 〈V F 〉 by 〈V 〉 =

∑α〈Vα〉. This approximation

which only uses the averaged potential is called the virtual crystal approximation (VCA). It iseasy to handle because it leads to a real potential, but has serious deficits for describing real dis-ordered systems. A major improvement on the VCA is to use the first term of (53) or (52). Thisapproximation is called average tmatrix approximation (ATA) and takes into account all effectsup to first order in the t operator. The ATA has been extensively discussed in the literature, inparticular in connection with multiple-scattering theory [48, 49]. A simple way to go beyondthe ATA is to use all terms of (53) which contain only a single site α. The result can writtenas∑

α(1 + g〈tα〉)−1〈tα〉 and corresponds to the optical potential discussed by Goldberger andWatson [51]. This method is correct to second order in the t operator, the third order terms nottreated are the last two ones shown in (53). An even better approximation is obtained by real-izing that the above equations are valid for quite arbitrary Green functions g provided that thecorresponding reference system shows no disorder. In a self-consistent procedure the referenceGreen function g is chosen in such a way that as many terms as possible vanish in (53). In prin-ciple, one could demand 〈tα(r, r′)〉 = 0. In general this is, however, not possible because theintegral operator tα(r, r′) is a complicated function of r and r′. In the multiple-scattering KKRmethod the situation is somewhat simpler because it relies on algebraic matrices tn instead ofintegral operators tα. Then, as shown by Soven [50] self-consistency can be implemented bychoosing an effective reference medium with the requirement that the scattering of the electronsaveraged over the constituent atoms vanishes in the effective medium. The resulting approxi-mation called KKR-CPA has been widely applied to study the electronic structure of disorderedalloys.As an example for extensive KKR-CPA calculations Fig. 11 shows results [52] for the averagedlocal magnetic moments in alloys of Fe, Co, and Ni and alloys of these elements with othertransition metals. The diagrams obtained on the left for the experimental results and on theright for the calculated results are referred to as Slater-Pauling curves. The Slater-Pauling curvehas two main branches with slopes of 45o and -45o which meet in the middle where a maximal

10.22 Rudolf Zeller

Fig. 11: Slater-Pauling curve for the averaged magnetic moment per atom as function of theaveraged number of electrons per atom. The picture is taken from Ref. [52].

moment of about 2.4 µB occurs. The left main branch consists of Fe alloys, whereas Co and Nialloys form the right main branch and the subbranches. The main reason for the two differentslopes is a different electronic screening behavior. Alloys on the main branch on the right have afull majority spin band so that the screening of the valence difference introduced by the impurityatoms is provided by minority spin electrons. This leads to a reduced number of minority d-electrons which gives increased moments. Alloys on the other branches are characterized by theoccurrence of antiparallel moments of the impurities which lead to reduced averaged momentswith increasing concentration. Here the screening is mainly provided by the majority spinelectrons. Although the agreement between experiment and calculation is not perfect, Fig. 11shows that spin density functional theory is a powerful tool to understand and explain magneticproperties of materials.


Appendix

A Useful Green function properties

The Green function G(ε) for a HamiltonianH is defined by the operator equation

G =1

ε−H. (54)

For real ε it is necessary to perform a limiting process. Then the real quantity ε is replaced bya complex quantity ε + iγ and all equations are understood in the sense that the limit γ → 0+

must be performed in the end. The relation

limγ→0+

1

ε+ iγ −H= P

1

ε−H− iπδ(ε−H) (55)

where P denotes the principal value, establishes the connection between the imaginary part ofthe Green function and the density of states n(ε) = δ(ε − H) and also the Kramers-Kronigrelation between the imaginary and the real part of the Green function

ImG(ε) = −πn(ε) , ReG(ε) = − 1

πP

∫ ∞−∞

1

ε− ε′ImG(ε′) dε′ . (56)

where the last equation is a Hilbert transform.While the above equations in this appendix are valid for real values of ε, complex values of εare considered below. Then the last equation can be generalized to

G(ε) = − 1

π

∫ ∞−∞

1

ε− ε′ImG(ε′) dε′ . (57)

Another useful relation for the Green function is given by

dG(ε)

dε=

d

dε

1

ε−H= − 1

ε−H1

ε−H= −G(ε)G(ε) . (58)

The electronic density of states can formally be expressed as

n(ε) = − 1

πIm TrG (ε) . (59)

The difference between the density of states for two systems characterized by two Green-function operators G(ε) and g(ε) is given by

∆n(ε) = − 1

πImTr[G(ε)−g(ε)] = − 1

πImTr[g(ε)V G(ε)] = − 1

πImTr

[g(ε)V

1

1− g(ε)Vg(ε)

].

Here and below V denotes the difference between the two potentials. By use of (58) for g(ε)this can be expressed as

∆n(ε) =1

πImTr

[dg(ε)

dεV

1

1− g(ε)V

]= − 1

πImTr

d

dεln(1− g(ε)V ) . (60)

The difference between the integrated densities of states is thus given by

∆N(ε) =

∫ ε

dε′∆n(ε′) = − 1

πImTr ln(1− g(ε)V ) (61)

10.24 Rudolf Zeller

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